# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: * **Finite Geometric Series:** The sum of a specific number of terms in a geometric sequence. This is used when the series has a defined end. * **Infinite Geometric Series:** The sum of all terms in a geometric sequence that continues indefinitely. * **Convergent Series:** An infinite geometric series that has a finite sum. This occurs only when the absolute value of the common ratio *r* is less than 1 (i.e., $|r| < 1$). The word problems and repeating decimal questions rely on this concept. * **Repeating Decimals as Series:** A repeating decimal can be expressed as an infinite geometric series, allowing it to be converted into a fraction. For example, $0.444... = 0.4 + 0.04 + 0.004 + ...$ Understanding these concepts is crucial for solving the exercise questions effectively. --- # **Important Formulas** Below are the key formulas used in this exercise: | Formula | Description | | :-------------------------------------- | :-------------------------------------------------------------------------------------- | | **Sum of a Finite Geometric Series** | The formula to find the sum ($S_n$) of the first *n* terms of a geometric series. | | $S_n = \frac{a_1(1 - r^n)}{1 - r}$ | Where $a_1$ is the first term, *n* is the number of terms, and *r* is the common ratio. | | **Sum of an Infinite Geometric Series** | The formula to find the sum ($S$) of a convergent infinite geometric series. | | $S = \frac{a_1}{1 - r}$ | This formula is only valid when the common ratio *r* is between -1 and 1 (i.e., $r<1$). | --- # **Summary** This exercise provides a comprehensive exploration of geometric series, covering both finite and infinite cases. The initial questions focus on calculating the sum of a finite number of terms using the standard formula. The exercise then transitions to problems where one must work backward to find the first term when the sum is known. A significant part of the exercise is dedicated to the application of infinite geometric series, specifically in converting repeating decimals to fractions and solving real-world problems like calculating the total distance traveled by a bouncing ball or the maximum height of a balloon. The key challenge lies in identifying whether a series is finite or infinite and applying the correct formula accordingly. --- # **Reference Videos** Below are recommended video explanations for this exercise: